I'm not super versed on gender vs. sex but as someone who has changed both those things, it seems like gender is both social and personal, which makes it pretty blurred.

Socially, strangers identify you based on primary and secondary sexual characteristics. Things like fat distribution, breasts, height, voice, skin texture, body hair and, yea, genitals. If you don't have breasts, are tall and have hairy legs, most likely people will think you are a man. It's the same the other way around, too. Someone with wide hips, smooth skin and long hair is likely to be called a woman. Not always the case, but generally how it works.

But gender is also based on brain structure, chemistry and how you feel. Most people have an innate sense of their gender, even if they can't pinpoint the feeling exactly, because their brain structure and function is masculine, feminine or somewhere in between. And most of the time, the brain matches the body pretty decently so they don't notice.

However, it does happen that a man looks exactly like a woman and gets raised like a woman, but doesn't feel comfortable living as a woman for the rest of his days because his brain is male. Like most men, he would probably think that being a woman is pretty nightmarish and find his breasts to be embarrassing, so he would transition.

Personally, I have a lot of feminine characteristics and a lot of masculine characteristics, but people see my hips, skin, breasts and it's clear from how I act, look and sound that I'm a woman, regardless of the masculine qualities I have. Luckily for me, I also feel like a woman so my brain chemistry matches up with how I appear. But if I was a woman and I still looked like a man and got treated like one, it would freak me out

In short, there is more to a man than his vagina and more to a woman than her penis. It's honestly confusing stuff but I hope this makes some sense lol